A solution of the integral equation for an oscillating, two-dimensional, thin airfoil in a compressible flow (subsonic and inviscid) is obtained by retaining only first order terms in frequency. The results are applied to the calculation of the damping derivative of a tail in rotary motion about a forward center, and it is shown that the damping is considerably less than that calculated on the basis of stationary airfoil theory. A brief investigation of induction effects shows this reduction to be considerably less for a wing of finite aspect ratio.